Robustness and resilience in complex systems
Unraveling how networks respond to disturbances and why it's important for our future
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Dall-e 3 representation of this issue’s content
Complexity and robustness
Broadly speaking, robustness is the ability of a system to maintain its function(s) in the face of internal and external disturbances. However, not all systems are the same.
In their seminal paper, Carlson and Doyle argue that system’s robustness is strictly related to its complexity, where the latter arises not just from the number of components or interactions but from the organized structures that enable these systems to adapt, evolve, and function in uncertain environments. In other works, complex systems self-organize to be as robust as possible, exactly to maintain their function.
There is a huge literature — too long to be exhaustively cited — about robustness of complex systems in biology, physics, engineering and social sciences. We can argue that robustness to perturbations is an essential feature of complex systems and for obvious reasons. If the Internet was not robust enough to random and temporary disconnection of the vast majority of its autonomous systems then it would be nearly impossible to use online services. If cells were not robust enough to mutations or copy errors in the DNA then it would be impossible for an organism to reproduce and evolve (see this insightful book by Andreas Wagner or this work focused on regulatory networks). If ecosystems were not stable enough to environmental changes then they would collapse (this work, together with previous studies links robustness to dynamical stability).
What can be captured by networks?
Among the variety of complex systems out there, many exhibit intricate connectivity patterns among their units, from cells in a biological system, to individuals in a community. Such interactions form the backbone of what we usually understand as complex networks: they are not just mere collections of nodes and links; they are dynamic structures woven with patterns of heterogeneity, modularity, and hierarchy reflecting some (not completely understood, yet) principles of organization and function in various systems. This diversity in structure has deep implications for how information is processed, how resilience is built, and how systems adapt and evolve.
Remarkably, more than two decades ago it has been shown that shocks due to internal failures or targeted attacks to specific units could hinder the function of a network, as measured by the integrity of its connectivity (eg., the relative size of the largest connected component). This is based on the hypothesis that only interconnected nodes can exchange information and operate together to allow for system function.
This is an interesting assumption, but it is not the whole story. In fact, one can wonder if it is possible to impair system function without necessarily acting on its structure by disintegrating it, but just by altering information flow. According to a recent work from my lab the answer is YES, it is possible, opening the door to complementary analysis of system function in terms of structure and dynamics at the same time (click here for a ComplexityThoughts essay about this approach, based on the concept of density matrix).
Figure from Phys. Rev. Research.
But complex networks are even more sophisticated than that. In fact, the previous examples consider only static attacks to structure or information dynamics, while it is possible to show the existence of dynamic attacks where breaking down a single unit is enough to trigger a systemic collapse to hinder the function of the whole system.
Figure from Nature Reviews Physics.
Such attacks ignite cascade failures, like in a domino chain, showing how a system can significantly respond even to small, targeted, disturbances (see the video below) with low chances to avoid catastrophic consequences, especially when systems are interdependent from other systems (see the above figure).
Therefore, the robustness and the resilience of a complex network are related to its ability to withstand shocks such as internal failures or targeted attacks, as well as to cascade failures, be they sudden environmental changes, specific attacks on infrastructure, or the spread of a pathogen in a community or of misinformation in online social networks.
In an interconnected world, a small perturbation in one part can cascade through the network, leading to unforeseen and often significant consequences.
Understanding the robustness of these networks is a necessity for predicting and mitigating potential crises. A core concept to understand network robustness is phase transition, borrowed from physics but equally illuminating in the context of complex (non-physical) networks. These transitions, marked by abrupt changes in a system's qualitative behavior or structure, depends on variety of factors, from connectivity patterns to spatial embedding (e.g., how the physical layout of nodes affects interaction), interdependence (how networks rely on one another), and coupled dynamics (how networks influence each other over time).
Figure from Nature Reviews Physics.
Thanks to network science, statistical physics and machine learning, we have a rich theoretical framework and computational tools to dissect these aspects of complex systems, that can be used to predict system behavior via simulations that test the impact of hypothetical scenarios. The additional advantage of these methodologies is that they do not only allow us to quantify the resilience of systems: they also offer insights into designing networks that can better withstand perturbations, identify early-warning signals of impending transitions, and devise strategies for adaptive and robust responses to threats.
Applications are broad
The applications of these insights are as varied as the fields that complexity science touches. In systems biology, they help in understanding how cellular networks respond to genetic mutations or environmental stressors. In neuroscience, they shed light on how brain networks reorganize in response to injury or disease. Engineering benefits from designing more resilient infrastructure, while insights into social and behavioral networks can inform policies to prevent the spread of misinformation or understand the dynamics of social movements.
I have discussed with some colleagues about applications in specific domains.
Relevance for physical systems
I have asked Prof. Shlomo Havlin (Bar-Ilan U.) to comment about the importance of this research field for physics:
We developed the theoretical framework for the robustness of interdependent networks, realizing that dependency interactions yield fascinating novel critical phenomena that occur in percolation, such as abrupt transitions due to microscopic cascading back and force between the networks. This rich phenomena led us to hypothesize that such transitions could be observed in physical systems. I approached in 2011 the experimentalist, A. Frydman, working in superconductivity of a single network layer, suggesting to generate and perform experiments on two interdependent superconducting layers. Indeed, the experimental results recently published in Nature Physics proved the theory of interdependent networks.
Figure from Nature Physics.
Relevance for biological systems
Living organisms are robust since they can maintain performance under a broad range of random perturbations, ranging from temporary chemical or physical changes in the environment, to permanent genetic mutations — Aldana et al.
Biology is extensive and from regulatory genetic networks to whole organisms, there are several case studies where biological robustness plays a crucial role (see also this paper).
I have asked Dr. Giulia Menichetti (Harvard Medical School) to provide me with a comment from the perspective of network medicine:
Network Medicine relies on the human interactome or protein-protein interaction network (PPI), an extensive subcellular map consisting of all known physical and regulatory interactions between human proteins and RNAs, leveraged for understanding disease mechanisms, drug discovery, and repurposing.
The structure of the underlying interactome plays an essential role in the system’s ability to efficiently propagate biological signals and survive random failures (e.g., countless misfolding errors or missed reactions), displaying the typical behavior of the universal scale-free topology for error/attack tolerance and emergence of cascading failure. Overall, the several layers of complex molecular interactions embedded in the interactome ensure:1) Robustness. On average, the cellular network maintains its basic functions in the presence of internal and external errors.
2) Resilience. On average, the cellular network adapts to internal and external errors by changing its mode of operation.
3) Redundancy. The cellular network displays parallel components and functions (e.g., different pathways, protein families) that, if needed, can replace a missing element.The majority of the analyses in Network Medicine revolve around quantifying the efficiency of the communication between different regions of the interactome, and to what degree these observations are expected. Most commonly, the speed and reliability of the exchanged signals are quantified through shortest-path or diffusion-based metrics, while the expectations are set by degree-matching or degree-preserving null models to highlight biological properties which are not solely determined by the degree distribution. For example, proteins involved in similar therapeutic areas or disease modules are expected to create a cohesive functional subgraph of proteins effectively communicating and influencing each other. Similarly, the areas of the interactome perturbed by a drug or a food compound should be proximal to its protein targets. Network incompleteness or annotation bias can affect these predictions comparably to random node/link removal or attack.
In a brief exchange, Prof. Luiz Pessoa (U. of Maryland) provided me with a perspective from neuroscience:
While robustness and resilience are important properties of complex systems, we know very little about how these properties are realized in the brain. They might involve redundancy and parallel processing, and/or other important topological properties of large-scale networks, among several other possibilities. Others might disagree, but I don't feel we really know much about it.
Note: This is an interesting perspective, and most likely I will try to dig into it with Luiz and other neuroscientists in a future post.
Relevance for urban systems
I have asked Prof. Marta Gonzalez (UC Berkeley) to discuss with me about the relevance of robustness and resilience studies for urban systems:
Cities are marked by a high concentration of people living in a relatively small area. This density is often due to economic opportunities, better infrastructure, and access to services. In this context, shared resources can be overused or depleted because individuals act in their self-interest. Roads and highways are used by numerous individuals trying to minimize their traveled distance. Also peak demand often occurs during periods of high electricity usage, such as late afternoons and early evenings when people return home from work, and use household appliances or charge their cars. In these systems, network science allows to map the demand of the infrastructures (roads, power lines, etc.) vs. their existing capacity and their interdependencies. By identifying the critical load rates we can predict the points of collapse of the system and plan for demand management strategies or recovery plans. Identifying the vulnerable elements is the first step to maintain network functionality to be more resilient to disruptions, or failures.
Relevance for social systems
I have asked Prof. Alex Arenas (U. Rovira i Virgili) to share with us his take on how network robustness and resilience influence social dynamics, such as epidemics:
The analysis of robustness and resilience in social systems poses significant challenges due to the profound impact of social behavior. For instance, in the context of epidemics, fluctuations in behavioral dynamics—shaped by enforcement measures, public opinions, climate conditions, and other factors—can be extreme. Such variability compromises the system's resilience to adapt to phenomena and its robustness against the collapse of essential services. Consequently, the necessity of delving deeply into robustness and resilience from a network science perspective becomes paramount in a world where behaviors change at an unprecedented pace, driven by the hyper-communication characteristic of our societies. This approach is crucial for understanding the complex interplay between social behaviors and systemic vulnerabilities, and for developing strategies to enhance the adaptability and sustainability of these systems amidst rapid global changes.
Preventing and reacting to network collapse
Figure from Nature Reviews Physics.
From ecosystems that sustain life to the communication and transportation infrastructures that sustain our societies, the ghost of network collapse looms large, with far-reaching consequences. This simple fact calls for an important area of study to investigate how to prevent or mitigate network collapse.
In fact, resilience is not inherent: it requires deliberate design choices and strategic interventions. For instance, in ecological networks, biodiversity can act as a buffer against environmental stresses, suggesting that preservation efforts can enhance ecosystem resilience. Similarly, in engineered systems, redundancy and decentralization are key factors to ensuring that the failure of a single component does not trigger (or can trigger with low probability) a system breakdown.
To this aim, simulating a spectrum of failure scenarios allows us to identify critical units and connections, as well as to understand the conditions under which they might lead to collapse. This predictive capability is vital for taking action before it is too late, enabling policymakers to reinforce or reconfigure critical networks to mitigate risks or damages.
When prevention is not enough, reacting to network collapse requires a different set of strategies: rapid response mechanisms, built on real-time monitoring and data analysis, are crucial. These systems can trigger some alerts at the first signs of trouble: early-warning detection of such signs is crucial for preventing ecosystems regime shift, pandemics or infrastructures breakdown.
The key is agility: the faster the response, the more contained the damage.
Figure: behavior of a systemic indicator over time, highlighting how robustness and resilience change with the system response. Example from PLOS Global Public Health.
Equally important is the capacity for recovery and learning from collapse. Post-crisis analyses provide invaluable insights into network vulnerabilities and the effectiveness of response strategies. This learning process feeds back into both prevention and reaction plans, ensuring that with each challenge, the network becomes more robust. For instance, after a financial system crisis or a pandemic, regulators and institutions often implement new safeguards and monitoring systems to prevent future collapses.
Figure: illustration of different routes to recovery over time, which depends on the pre-existing features of a system at the time when a shock hits (eg, the COVID-19 pandemic) as well as on the interventions adopted to mitigate the spread of disruptions. Example from Bulletin of the WHO.
Studying network collapse, together with prevention and reaction strategies, requires a confluence of expertise from systems biology, computer science, ecology, engineering, and social sciences, among others. Each discipline contributes a piece of the puzzle, from understanding the natural laws (here, I refer to the fundamental insights provided by statistical physics, among others) that govern network behaviors to designing technological and policy interventions.
Nevertheless, preventing and reacting to network collapse is an ongoing challenge — and, most likely, it will still be in the future — reflecting the dynamic and evolving nature of complex systems. The strategies developed today may need to be reevaluated tomorrow, as networks grow, change and adapt.
Yet, this challenges also present an opportunity: to innovate, to strengthen the interconnectivity that underpins modern life, and to ensure that our systems, like the societies they support, are resilient, adaptive, and sustainable.
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